Unplug the landscape lights.
Peel back 1 inch of the outer insulation on both sides of the severed wire with a utility knife, carefully exposing the two wires inside.
Strip 1/2 inch of the insulation away from the two inside wires in each of the sides, exposing the metal wire inside.
Twist the severed wires together to form a complete circuit. With low voltage wiring it doesn't matter which wire is joined to which.
Wrap the splicing kit around the splice and snap it together. The splice is now protected against moisture and dirt.